Constructive proof demonstrates smooth projective hypersurfaces violating modular point congruence over prime fields, highlighting boundaries of classical polynomial theorems.
For every prime p, integer n >= 2, and degree d >= n+1, we construct a geometrically smooth hypersurface X in projective n-space over F_p of degree d such that #X(F_p) is not congruent to 1 modulo p. The construction stays over the specified prime field in every characteristic and degree. A finite-field moment polynomial supplies nonzero coefficients with the required point count. In odd characteristic not dividing the degree, a sharper individual-degree bound permits simultaneous avoidance of the singular parameter. When the characteristic divides the degree, positive exponent compositions make a triangular family smooth for all nonzero coefficients, apart from one explicitly handled boundary case. A separate uniform formula treats characteristic two in odd degree. The coefficient selection is a finite deterministic procedure; no efficient complexity bound is asserted. This is an unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported literature search, computation, proof auditing and manuscript preparation. The author remains responsible for the claims and final text. No absolute priority claim is made. Corpus identifier: AIM-ALGEBRAIC_GEOMETRY-0125. EulerSolve page: https://eulersolve.org/papers/aim-algebraic-geometry-0125/
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Alper Ferudun (2026) studied this question.